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    The fact that toposes support closed set logic as readily... — Carmelics
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    Home/Modality & Possibility
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    The fact that toposes support closed set logic as readily as open set logic is an argument that inconsistent theories are equally reasonable as items of mathematical study.

    Modality & PossibilityTruth & Knowledge
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    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Toposes support open set logic, which has been taken as a vindication of mathematical intuitionism.
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    • 2.It can be proved that toposes support closed set logic as readily as they support open set logic.
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    • 3.Closed set logic provides the only category-theoretic semantics for a paraconsistent logic to date.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The vindication of intuitionism from topos theory rests not merely on open set logic being supported, but on its deep connection to constructive proof and epistemic constraints on mathematical truth.
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    • 2.Closed set logic's category-theoretic availability does not supply paraconsistent mathematics with a corresponding epistemic or proof-theoretic motivation analogous to intuitionism's rejection of the law of excluded middle.
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    • 3.Structural parity in a semantic framework is insufficient for philosophical parity without an independent account of what cognitive or mathematical practice the logic is answerable to.
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    Reason against 2 of 2
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    • 1.Quine and Putnam's indispensability arguments ground mathematical ontology in the explanatory and predictive success of scientific theories, not in the mere formal consistency or categoricity of logical systems.
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    • 2.Inconsistent theories, lacking application in empirically successful science, fail the indispensability criterion that historically justifies treating classical and even intuitionistic mathematics as legitimate objects of study.
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    • 3.Category-theoretic generality that encompasses a formalism does not constitute the kind of scientific indispensability that warrants treating inconsistent theories as equally reasonable mathematical objects.
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    Related

    Category-theoretic generality that encompasses a formalism does not constitute t...Closed set logic provides the only category-theoretic semantics for a paraconsis...Closed set logic's category-theoretic availability does not supply paraconsisten...Inconsistent theories, lacking application in empirically successful science, fa...
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    It can be proved that toposes support closed set logic as readily as they suppor...Quine and Putnam's indispensability arguments ground mathematical ontology in th...Structural parity in a semantic framework is insufficient for philosophical pari...The vindication of intuitionism from topos theory rests not merely on open set l...Toposes support open set logic, which has been taken as a vindication of mathema...

    Similar

    It can be proved that toposes support closed set logic as readily as t...92%The fact that toposes support closed set logic should not be viewed as...89%The result that toposes support closed set logic equally to open set l...85%Inconsistent theories arising from closed set logic are no less natura...85%

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    SEP: mathematics-inconsistent
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    Category theory throws light on many mathematical structures. It has certainly been proposed as an alternative foundation for mathematics. Such generality inevitably runs into problems similar to those of comprehension in set theory; see, e.g., Hatcher 1982 (pp. 255–260). Hence there is the same possible application of inconsistent solutions. There is also an important collection of categorial structures, the toposes, which support open set logic in exact parallel to the way sets support Boolean
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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit